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T1 Mapping · Exercise 1

Inversion recovery: null times and contrast

  1. a. Using the Plotly figure in the inversion recovery section that displays the signal curves for white matter, gray matter, and CSF, determine the inversion times that null the signal from each of these tissues.
  2. b. In practice, you may not have this type of interactive figure at the scanner. Using the three values nulling inversion time values above, can you find an easy way to approximate the nulling time for any arbitrary T1 value?
  3. c. Assuming that the images acquired at the MRI scanner displays magnitude-only images, at approximately which inversion time will the white matter and grey matter have the lowest contrast?
  4. d. Which inversion time would result in the best contrast between white matter and grey matter?

a. Find the null inversion times

a. Using the Plotly figure in the inversion recovery section that displays the signal curves for white matter, gray matter, and CSF, determine the inversion times that null the signal from each of these tissues.

Hover over each curve where it crosses zero, and type your readings here. They carry over to the next slides.

TissueNull TI (ms)
White matter
Grey matter
CSF

Figure 2.2 from the book. Its TI axis is in seconds: a reading of 0.62 s is 620 ms.

b. A rule of thumb for the null time

b. Using the three values nulling inversion time values above, can you find an easy way to approximate the nulling time for any arbitrary T1 value?

TissueT1 (ms)Null TI read from the figure (ms)
White matter
Grey matter
CSF

Setting Eq. 2.3 to zero:
$$\begin{aligned} 0 &= C\left(1-2e^{- \frac{TI_{null}}{T_1}}\right) \;\Rightarrow\; e^{- \frac{TI_{null}}{T_1}} = \frac{1}{2} \\ TI_{null} &= T_1 \ln 2 \approx 0.693\,T_1 \end{aligned}$$

c. Lowest WM/GM contrast in magnitude images

c. Assuming that the images acquired at the MRI scanner displays magnitude-only images, at approximately which inversion time will the white matter and grey matter have the lowest contrast?

Between the two nulls, |Mz| of WM and GM cross at TI ≈ 800 ms: the two tissues look the same.

The crossing solves e−TI/T1,WM + e−TI/T1,GM = 1, with the T1 values of Figure 2.2.

d. Best WM/GM contrast in magnitude images

d. Which inversion time would result in the best contrast between white matter and grey matter?

The magnitude contrast peaks at TI ≈ 1150 ms.

Both tissues are past their null there, so |Mz| = Mz. Setting the derivative of Eq. 2.3 for WM minus GM to zero:

$$\begin{aligned} &\frac{d}{dTI}\left(e^{-\frac{TI}{T_{1,GM}}} - e^{-\frac{TI}{T_{1,WM}}}\right) = 0 \\ &\Rightarrow\; TI_{best} = \frac{T_{1,WM}\,T_{1,GM}}{T_{1,GM}-T_{1,WM}}\ln\frac{T_{1,GM}}{T_{1,WM}} \end{aligned}$$

Exercise 1: answers

PartAnswer
aNull TIs ≈ 624 ms (WM), 1040 ms (GM), 2773 ms (CSF)
bTInull ≈ 0.69 T1 = T1 ln 2 (long-TR approximation)
cLowest magnitude contrast at TI ≈ 800 ms, where |Mz| of WM and GM cross
dBest contrast at TI ≈ 1150 ms

Discuss Exercise 1 on GitHub: github.com/qMRLab/mooc/discussions/4

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